| Abstract: | We consider the equation in question on the interval 0 ≦ x ≦ 1 having Neumann boundary conditions, with f(u) = F(u), where F is a double well energy density with equal minima at u = ±1. The only stable states of the system are patternless constant solutions. But given two-phase initial data, a pattern of interfacial layers typically forms far out of equilibrium. The ensuing nonlinear relaxation process is extremely slow: patterns persist for exponentially long times proportional to exp{A±l/?, where A = F(±1) and l is the minimum distance between layers. Physically, a tiny potential jump across a layer drives its motion. We prove the existence and persistence of these metastable patterns, and characterise accurately the equations governing their motion. The point of view is reminiscent of center manifold theory: a manifold parametrising slowly evolving states is introduced, a neighbourhood is shown to be normally attracting, and the parallel flow is characterised to high relative accuracy. Proofs involve a detailed study of the Dirichlet problem, spectral gap analysis, and energy estimates. |
